Families of noncongruent numbers
Number Theory
2015-06-26 v1
Abstract
Let E_k denote the elliptic curve defined by y^2 = x(x^2 - k^2). We consider the curves with k = pl, p = l = 1 mod 8 primes, and show that the density of rank-0 curves among them is at least 1/2 by explicitly constructing nontrivial elements in the 2-part of the Tate-Shafarevich group of E_k.
Cite
@article{arxiv.math/0202311,
title = {Families of noncongruent numbers},
author = {Franz Lemmermeyer},
journal= {arXiv preprint arXiv:math/0202311},
year = {2015}
}